Asymptotic distribution of a Pickands-type estimator of the extreme-value index

Laurent Gardes; Stéphane Girard

doi:10.1016/j.crma.2005.05.023

Statistics/Probability Theory

Asymptotic distribution of a Pickands-type estimator of the extreme-value index

Presented by: Paul Deheuvels

Laurent Gardes ¹; Stéphane Girard ²

¹ Université Grenoble 2, LABSAD, B.P. 47, 38040 Grenoble cedex 9, France
² Université Grenoble 1, LMC-IMAG, B.P. 53, 38041 Grenoble cedex 9, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 1, pp. 53-58

Abstract (Original language)
Résumé

One of the main goals of extreme-value analysis is to estimate the probability of rare events given a sample from an unknown distribution. The upper tail behavior of this distribution is described by the extreme-value index ξ. The aim of this Note is to establish the asymptotic distribution of the estimator of $ξ \in R$ introduced in Gardes and Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004]. We also give its rate of convergence in some typical situations.

Dans de nombreuses applications, on s'intéresse à l'estimation de probabilités d'évènements rares étant donné un échantillon d'une loi inconnue. Pour ce faire, il est généralement indispensable d'estimer au préalable l'indice des valeurs extrêmes ξ qui caractérise le type de décroissance de la queue de distribution. L'objectif de cette Note est de déterminer la loi asymptotique de l'estimateur de $ξ \in R$ défini dans Gardes et Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004] et d'en déduire sa vitesse de convergence pour certains modèles.

Received: 2005-01-18
Accepted: 2005-05-24
Published online: 2005-06-29

DOI: 10.1016/j.crma.2005.05.023

Author's affiliations:

Laurent Gardes ¹; Stéphane Girard ²

¹ Université Grenoble 2, LABSAD, B.P. 47, 38040 Grenoble cedex 9, France
² Université Grenoble 1, LMC-IMAG, B.P. 53, 38041 Grenoble cedex 9, France

@article{CRMATH_2005__341_1_53_0,
     author = {Laurent Gardes and St\'ephane Girard},
     title = {Asymptotic distribution of a {Pickands-type} estimator of the extreme-value index},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {53--58},
     year = {2005},
     publisher = {Elsevier},
     volume = {341},
     number = {1},
     doi = {10.1016/j.crma.2005.05.023},
     language = {en},
}

TY  - JOUR
AU  - Laurent Gardes
AU  - Stéphane Girard
TI  - Asymptotic distribution of a Pickands-type estimator of the extreme-value index
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 53
EP  - 58
VL  - 341
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2005.05.023
LA  - en
ID  - CRMATH_2005__341_1_53_0
ER  -

%0 Journal Article
%A Laurent Gardes
%A Stéphane Girard
%T Asymptotic distribution of a Pickands-type estimator of the extreme-value index
%J Comptes Rendus. Mathématique
%D 2005
%P 53-58
%V 341
%N 1
%I Elsevier
%R 10.1016/j.crma.2005.05.023
%G en
%F CRMATH_2005__341_1_53_0

Laurent Gardes; Stéphane Girard. Asymptotic distribution of a Pickands-type estimator of the extreme-value index. Comptes Rendus. Mathématique, Volume 341 (2005) no. 1, pp. 53-58. doi: 10.1016/j.crma.2005.05.023

References
Cited by

[1] N.H. Bingham; C.M. Goldie; J.L. Teugels Regular Variation, Cambridge University Press, 1987

[2] L. Gardes, Estimation d'une fonction quantile extrême, PhD Thesis, Université Montpellier 2, 2003

[3] L. Gardes, S. Girard, A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004

[4] B. Gnedenko Sur la distribution limite du terme maximum d'une série aléatoire, Ann. Math., Volume 44 (1943), pp. 423-453

[5] P. Hall On some simple estimates of an exponent of regular variation, J. Roy. Statist. Soc. Ser. B, Volume 44 (1982), pp. 37-42

[6] J. Pickands Statistical inference using extreme-order statistics, Ann. Statist., Volume 3 (1975), pp. 119-131

Cited by Sources:

Comments - Policy